SEPARATION OF THE «-SPHERE BY AN (n - 1)-SPHERE BY JAMES C. CANTRELL(i)
1. Introduction. Let A be the closed spherical ball in E" centered at the origin 0, and with radius one, B the closed ball centered at O with radius one-half, and C the closed ball centered at O with radius two. The Generalized Schoenflies Theorem states that, if h is a homeomorphism of Cl (C —B) into S", then h(BdA) is tame in S" (the closure of either component of S" —h(BdA) is a closed n-cell) [5]. One is naturally led to the following question : if h is a homeomorphism of C\(A—B) into S", is the closure of the component of S" — h(BdA) which contains n(BdB) a closed n-cell? This question is answered affirmatively by Theorem 1 and should be listed as a corollary to the Generalized Schoenflies Theorem. Let D be the closed ball in E", centered at (0,0,-",0, — 1) with radius two. Two other types of embeddings of Bd.4 in S", n > 3, are considered in §2, (1) the embedding homeomorphism h can be extended to a homeomorphism of Cl(£>— B) into S" such that the extension is semi-linear on each finite polyhedron in the open annulus Int (A — B), and (2) h can be extended to a homeomorphism of C1(B —A) into S" such that the extension is semi-linear in a deleted neigh- borhood of (0, 0, ■■■,(), 1) (see Definition 1). Theorem 4 strongly suggests that, for an embedding of type (1), h(BdA) is tame in S". An embedding of this type corresponds to the three dimensional case in which h(BdA) is locally polyhedral except at one point. In §3, three methods of constructing 3-spheres in S4 from 2-spheres in S3 are considered: (1) suspension of a 2-sphere in S3, (2) rotation of a 2-cell in S3 about the plane of its boundary, and (3) capping a cylinder over a 2-sphere in S3. The construction methods in cases (1) and (2) were introduced by Artin [2] and have been used by him and by Andrews and Curtis [1] to construct 2-spheres in S4 from 1-spheres in S3. Their techniques may be applied directly to establish isomorphism theorems relating the fundamental groups of the complements of the constructed 3-spheres and the fundamental groups of the corresponding complements of the given 2-spheres. Thus, methods (1) and (2) may be used to construct wild (nontame) 3-spheres in S4. Method (2) is also used to construct
Presented to the Society, November 17,1961 under the title Some embeddings ofS"-1 in S"; received by the editors July 24, 1962. 0) The author wishes to express his appreciation to Professor O. G. Harrold, Jr., for directing the thesis, the principal results of which constitute this paper. This research was supported in part through the NSF-G8239. 185
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a 3-sphere in S4, one complementary domain of which is simply connected but is not an open 4-cell. The third method is used to construct a 3-sphere in S4 such that one complementary domain has a closure which is a closed 4-cell, and the other complementary domain is an open 4-cell but its closure is not a closed 4-cell.
2. Some embeddings of S"_1inS". The reader is referred to [5] for the definitions of inverse set and cellular set.
Theorem 1. Let h be a homeomorphism ofC\(A - B) into S" and let G be the component of S" — h(BdA) which contains h(BdB). Then C1G is a closed n-cell Proof. Let G' be the component of S" - h(BdB) which does not contain h(BdA). We first observe that C1G' is a cellular subset of G. For, if B¡ is the closed ballin£", centered at O with radius 1/2 + l/(i+2), i = l,2,---,and Gt is the component of Sn— /t(BdB¡) which contains G', then, by the Generalized Schoen- flies Theorem, C1G¡ is a closed n-cell. Furthermore ClGi+1 c G. and p|7=1ClG, = ClG'. Let g be a continuous mapping of C1L4 — B) onto A such that Bdß is the only inverse set. Define a mapping/ of C1G onto A by the equations f(x) = gh~\x), if xeClG-G', f(x) = g(BdB), if xeG'. The mapping / carries C1G continuously onto A such that the only inverse set is the cellular subset C1G' of G. Thus, by Theorem 2 of [5], C1Gis a closed n-cell.
Theorem 2. Let h be a homeomorphism ofCl(D — B) into S" and G be let the component of S" — h(BdA) which intersects h(BdD). Then G is an open n-cell. Proof. Let H be the component of S" - h(BdA) which contains h(BdB). By Theorem 1, CIH is a closed n-cell and, hence, there is a homeomorphism/ of A onto Cl// such that/ and n agree on BdA. Define a homeomorphism
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The mapping \¡j carries S" onto S", leaves p fixed, and has CIH as the only inverse set. Hence, G is carried homeomorphically onto S" — p, and is an open n-cell. Let By be the closed ball in E" which is centered at O and has radius three- fourths, and let L' be the closed segment of the x„-axis from (0,0, •••,0,3/4) to (0,0, -.0,1). Theorem 3. Let h be a homeomorphism ofC\(D — B) into S" and denote h(L') by Land «(0,0, "-,0,1) by p. Let G be the component of S"—h(BdA) which intersects n(BdD) and let H be the component of S" — h(BdB¡) which contains h(BdA). Then C\H is a closed n-cell and (C1G) —p is topologically equivalent to CIH - L. Proof. That CIH is a closed n-cell follows immediately from Theorem 1. Let K be the component of S" — h(BdD) which does not intersect h(BdA) and let g be a continuous mapping of C1(D — By) onto C1(D — A) such that (1) g is fixed on BdD, (2) g(BdBy) = BdA, and (3) L is the only inverse set under g. The mapping / of CIH onto C1G defined by f(x) = x, if Ex e K,ï f(x) = hgh~\x), if x eClH - K, is a continuous mapping of CIH onto C1G such that the only inverse set is Land f(L) = p. Hence, / is a homeomorphism of CIH - L onto C1G — p. If in Theorem 3 there exists a continuous mapping k of CIH onto CIH such that L is the only inverse set, then we can state that C1G is a closed n-cell. In fact, the product mapping kf1 is a homeomorphism of C1G onto CIH. Let us now suppose that n > 3 and that h is semi-linear on each finite polyhedron of Int (A —B) (we assume a curved decomposition of E" in which A, B, By, and L are polyhedra). Then h(BdBy) is a polyhedron and L is locally polyhedral except at p. Let s > 0 be such that S(s,p) c H and use Lemma 2 of [6] to obtain a homeomorphism
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is a continuous mapping of Cl// onto OH such that L is the only inverse set. Thus, we have the following theorem.
Theorem 4. Let n > 3 and let h be a homeomorphism of Cl(D — B) into S". If h is semi-linear on each finite polyhedron o/Int (A — B), then h{BdA) is tame in S".
The semi-linear condition in Theorem 4 is used only to shrink Lto a boundary point of Cl//. It seems that one should be able to remove this condition and retain the conclusion, since the local embedding at each point t of L, different from p, is as "nice" as the local embedding of an interval at one of its points. In fact, for each / e L, different from p one can find a homeomorphism ht of S" onto itself such that the subarc Lt of L from q to t is carried onto a linear segment. Definition 1. Let h be a homeomorphism of C1(D —A) into S". If there exists a neighborhood N of (0,0,-■•,0,1) in E" such that h is semi-linear on each finite polyhedron of Int (D —A) nJV, then we say that h is semi-linear on a deleted neighborhood of (0,0, ••■,0,1).
Theorem 5. Let n > 3 and h a homeomorphism of Cl (D - A) into S" such that h is semi-linear on a deleted neighborhood of (0,0, ■•■,0,1). If G is the component of S" —h(BdA) which intersects h(BdD), then C1G is a closed n-cell. Proof. The technique of proof used here is that used by Mazur in [8]. Let £>! be a cell, obtained from D by a slight contraction on E" toward (0,0, "-,0,1), such that (BdDj) - (0,0,---,0,l) is contained in D -A. Let Gr and G2, respectively, be the components of S"-n(Bd/)1) and S"-n(BdD) which are contained in G. We now observe that ClGi is homeomorphic to C1G. For, if g is a homeomorphism of E" onto itself which is fixed on BdD and carries BdDy onto Bd^4,then the mapping
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removed. The cell obtained from P by attaching C1G! and CIH to the interior boundary spheres of P with n_1 will be denoted by P. Let F be the part of the solid unit ball in E" centered at (0,0, ■••,0,1,0), de- termined by x„ ^ 0. Let {&}£0 be a sequence of points in the intersection of the plane Xy = x2 = ••• = x„_2 = 0 and BdP such that, if q¡ = (0,0,••-, a(n-i)i> ani)>then a(n_1)0 = 2, an0 = 0, the a{n-i)¡ converge monotonically to zero, and ani > 0 for i > 0. We then section F into a countable number of n-cells by projecting the (n —2)-plane x„ = x„_! = 0 onto each of the q¡. The section determined by
Figure 1
Let (h¡ be a homeomorphism of P¡ onto P- which leaves w2i fixed and carries w2i_! onto w2i+l. Let i/f¡be a homeomorphism of P[ onto Pi+1 which leaves w2i+y fixed and carries w2i onto w2i+2. We identify Pt with P, with wx identified with BdDy and w2 identified with BdA. The sets ClGj and CIH are then sewn to P along Wyand w2, respectively, with n_1. The resulting n-cell is denoted by P\. The sets CIGy and CIH are then sewn into alternate holes bounded by w2¡+1 and w2i+2 by the attaching homeomorphisms <£;•" «Mi«-1 : BdG1->w2i+1, xj/i-il/^yh'1 : BdH->w2i+2.
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The sets thus obtained from the P¡ and P\ are denoted by P¡ and P[ and we set
Since 4>yis the identity on vv2,we can extend 4>i to a homeomorphism of Pt onto P', and conclude that P[ is also a closed n-cell. In a similar manner we extend i/í¡ to a homeomorphism of P¡ onto P¡+1 and extend <¡>¡to a homeomorphism of Pi onto P,'. It then follows that each P¡ and each P[ is a closed n-cell. We now observe that Ft is a closed n-cell. We map the boundary of C2;_ x \j C2i onto the boundary of Pt with the identity homeomorphism. Since C2i_! uC2j and P¡ axe n-cells, this homeomorphism between their boundaries can be extended to a homeomorphism between the cells. These extensions for i = 1,2,•••,yield a homeomorphism of F onto Ft. We next observe that Ft is a copy of Cl(r\ - A) with ClGt sewn along one of the boundary spheres. This can be established by showing that F„ with Gt removed from P1; is homeomorphic to F, with Int Ci removed. Let X be the identity mapping on Cl - Int C[ and on Bd(C2; UC2i+1), t = 1,2, ••-. Since C2i U C2i+, and P/are closed n-cells and X restricts to a homeomorphism between their boundaries, X can be extended over their interiors. These extensions over each of the C2i \jC2i+1 yield the desired homeomorphism. We have seen that F1 can first be viewed as a closed n-cell, and secondly as C1GX sewn into a boundary sphere of a copy of Cl(D1 - A). We previously observed that a set of the second type is equivalent to CIG^ Hence C1G1; or equivalently C1G, is a closed n-cell, and Theorem 5 is proved. If one were able to remove the semi-linear condition in Theorem 4, then the semi-linear condition in Theorem 5 could also be removed(2). In this general form Theorem 5 would imply that a wild (n - l)-sphere is S", n > 3, must be "knotted" at more than one point, and that such simple examples of wild spheres as the Fox-Artin examples [3] for n = 3 do not exist in the higher dimensional spaces.
3. Some 3-spheres in S*. Definition 2. In £4 we take coordinates x1,x2,x3,x4 and let E3 be described by x4 = 0. Let a = (0,0,0,1) and b = (0,0,0, - 1). For a set A in E3 the suspension of A in £4 is the join of A and a u b, and is denoted by Susp A. The proof of Theorem 1 of [1] may be used directly to prove the following theorem. Theorem 6. Let S be a 2-sphere in E3 and K = Susp S. Let Ax and A2 be the bounded and unbounded components of E3 - S respectively, and Bu B2 the corresponding components of £4 - K. Then the injection homomorphism ij : n^Afj-^n^Bj), j = 1,2, is an onto isomorphism. (2) Added in proof. After this paper was sent to press the author was able to remove the semi-linear conditions in Theorem s4 and 5. These results, together with certain generalizations, will appear in print at a later date.
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Let £3 = {(x1,x2,x3,0)££4|x3 ^ 0} and let P be the plane x3 = x4 = 0. For x = (x1,x2,x3,0) and 0 ^ t < 2n we set R,(x) = (xt.X2.X3 cos t, x3 sin t), and for a subset M of £3 we set R(M) = {R,(x) \ x e M, 0 1%t < 2n}. For a subset N of£4wesetB~1(iV) = {ye£3|B,(>')eN for some 0 ^ t < 2n). If M is a 2-cell in £+ such that M C\P = BdM = d, and D is the bounded component of P - d, then the proof of Theorem 3 of [1] may be used to establish the following theorem.
Theorem 7. Let Ay and A2 be the bounded and unbounded components, respectively, of El -(MUD) and let By, B2 be the corresponding components ofE*- R(M). Then ny(A¡)« ny(B¡),i = 1,2. In [3] there are examples of 2-spheres in S3 such that one complementary domain has a nontrivial fundamental group. Elementary modifications of these examples will give 2-spheres in S3 such that the fundamental group of either complementary domain is nontrivial. These examples, together with Theorem 6 or Theorem 7, give the existence of 3-spheres in S4 such that either one or both complementary domains have nontrivial fundamental groups. In passing, we observe one difference between the spheres Susp S and R(M). Associated with each exceptional point peS there will be an arc, Susp p, of exceptional points on Susp S, and for each exceptional point peM there will be a simple closed curve, R(p), of exceptional points on R(M). We now use the rotation of a disk about P to construct a 3-sphere in S4, one complementary domain of which is simply connected but is not an open 4-cell. Let us first embed the 2-sphere S, discussed as Example 3.2 in [3], in
Figure 2
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Figure 3
£+ as indicated in Figure 2. The sphere S is to intersect P in a 2-cell D and C1(S - D) is denoted by M. If Lis the arc described as Example 1.3 in [3], the proof in [3] that £3 —L is simply connected may be used directly to show that A2 (the exterior of S in £+) is simply connected. Hence, by Theorem 7, B2 (the exterior of R(M) in £4(S4)) is simply connected. The cross section M \jR„(M) of R(M) is shown in Figure 3. Let A2 denote the exterior of M U Rn(M) in E3. It is shown in [3, Example 1.3] that C0 cannot be contracted to a point in A'2 - [W\jRn(Wy\. This fact is now used to show that R(W) is contained in no closed 4-cell subset of B2 whose complement in B2 is simply connected. Hence, B2 is not an open 4-cell. Suppose that such a 4-cell J did exist. Choose the base point for computing 7t1(ß2 - J) in P and so close to d that there is a path c0 in (B2 - J) r\P which cannot be contracted to a point in A2 - \_W\jRK(W)~\. Let £ be a unit disk in £2 with boundary e, and let A be a continuous mapping of e onto c0. Since nx(B2 — J) is trivial, there exists an extension H of h which carries £ into B2 —J. We then follow H by R _1 and obtain a singular 2-cell, R~1H(E), in A2 - R-1(J) •which is hounded by c0. Since A2 —R~1(J)czA2 — W, we see that c0 can be contracted to a point in ^42 - If and hence in the larger set A'2 — \W\jRn(Wy]. This contradiction establishes the desired conclusion. We now describe a third method for constructing (n — l)-spheres in S" and refer to this method as capping a cylinder. In £" we again take coordinates xux2,---,xn and let £"_1 be described by xB = 0.
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Lemma 1. Let S be an(n —2)-sphere in £"_1 with the bounded and un- bounded components of £"_1 — S denoted by Ay and A2, respectively. If CL42 (compactified at infinity) is a closed (n —l)-cell, then {Sx[0, lJJuiCL^ x [1]} is a closed (n — l)-cell.
Proof. Let h be a homeomorphism of C1^42 onto a standard unit ball B in £"_1. Let Sy = BdB and let S2 be the sphere concentric with S¡ and with radius one-half. Then h~1(S2) is an (n — 2)-sphere in £"_1 and if C is the component of E""1 — h~1(S2) which contains Au then, by Theorem 1, C1C is a closed (n - 1)- cell. We now observe that C1C consists of a closed annulus with C\Ay sewn along one boundary component and is, therefore, a copy of {S x [0, l^ufCL^ x [1]}.
Theorem 8. Let S, Ay, and A2 be as in Lemma 1. If CIA2 (compactified at infinity) is a closed (n-l)-cell, then {Sx[-l,l]}\j{ClAy x [-l^ufCL^ x [1]} is an (n — l)-sphere.
Proof. By Lemma 1, each of {Sx[ - 1,0]} u{CUt x [ - 1]} and {S x [0, l^ufCl^! x [1]} is a closed n-cell. These cells meet along their common boundary sphere S, and hence their union is an (n — l)-sphere. We now consider a 2-sphere S, locally polyhedral except at a single point, in £3(S3) such that the bounded complementary domain Ay is an open 3-cell, ClAy is not a closed 3-cell, the unbounded complementary domain (compactified at infinity) is an open 3-cell, and CL42 is a closed 3-cell. The assertion is that the 3-sphere
r={Sx[-l,l]}u{CL41x[l]}u{CL41x[-l]}
is embedded in S4 such that, if By and B2, respectively, are the components of S4 — Twhich contain At and ^42, then By is an open 4-cell, ClBy is not a closed 4-cell, and C1B2 is a closed 4-cell. Since By is the product of the open 3-cell Ay and the open interval ( — 1,1), By is an open 4-cell. If ClBy =C\Ai x [- 1,1] were a closed 4-cell, a theorem due to Bing [4] would imply that C\A1 is a closed 3-cell. Thus contradicting our assumption on the embedding of S in S3. We now show that C1B2 is a closed 4-cell by constructing a homeomorphism / : Tx [0,1/2] -> C1B2such that/0(j;) =f(y, 0) = y for each y e Tand then applying Theorem 1. Since CL42 is a closed 3-cell, there exists a homeomorphism n : S x [0,1/2] -* CL42 such that h0(x) = h(x, 0) = x for each xeS. For yeT, let x be the point of CL4.J which lies under y (y=(x,f) for some ie[ - 1,1]). We define / by the following equations : (l)/,O0 = (x,l + r), ify = (x,l); (2)/r(y) = (x,-l-r), ify = (x,-l); (3) fr(y) = (K(x),t), if xeS and - 1 + r < t < 1 - r;
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 194 J. C. CANTRELL (4) fr(y) = (A(i-,)(*), 2i - (1 - r)), if x e S and 1 - r ^ í ^ 1 ; (5) /,G0 = (A(1-,)(x), 2« - (r - 1)), if x e S and - 1 ^ í ^ - 1 + r. The continuity off follows rather quickly from the definition off in terms of the continuous mapping h and a set of linear equations. The one-to-one property off depends principally on the fact that each arc/r(x x [0,1]) must lie over the arc Lx = {hs(x)\se [0,1/2]} and thatLXl and L,^ intersect if and only if x1=x2.
Bibliography 1. J. J. Andrews and M. L. Curtis, Knotted 2-spheres in the 4-sphere, Ann. of Math. (2) 70 (1959), 565-571. 2. E. Artin, Zur Istopie zweidimensionaler Flächen im R4., Abh. Math. Sem. Univ. Hamburg 4 (1925), 174-177. 3. E. Artin and R. H. Fox, Some wild cells and spheres in three dimensional space, Ann. of Math. (2) 49 (1948), 979-990. 4. R. H. Bing, A set is a 3-cell if its cartesian product with an arc is a 4-cell, Proc. Amer. Math. Soc. 12 (1961), 13-19. 5. M. Brown, A proof of the Generalized Schoenflies Theorem, Bull. Amer. Math. Soc. 66 (1960), 74-76. 6. J. C. Cantrell and C. H. Edwards, Almost locally polyhedral curves in Euclidean n-space, Trans. Amer. Math. Soc. 107 (1963),451-457. 7. V. K. A. M. Gugenheim, Piecewise linear isotopy and embedding of elements and spheres. I, Proc. London Math. Soc. 3 (1953), 29-53. 8. B. Mazur, On embeddings of spheres, Bull. Amer. Math. Soc. 65 (1959), 59-65. 9. E. C. Zeeman, Unknottingspheres, Ann. of Math. (2) 72 (1960), 350-361.
University of Tennessee, Knoxville, Tennessee
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